redist.compactness is used to compute different compactness statistics for a shapefile. It currently computes the Polsby-Popper, Schwartzberg score, Length-Width Ratio, Convex Hull score, Reock score, Boyce Clark Index, Fryer Holden score, Edges Removed number, and the log of the Spanning Trees.

distr_compactness(map, measure = "FracKept", .data = cur_plans(), ...)

redist.compactness(
shp = NULL,
plans,
measure = c("PolsbyPopper"),
total_pop = NULL,
draw = 1,
ncores = 1,
counties = NULL,
planarize = 3857,
ppRcpp,
perim_path,
perim_df
)

## Arguments

map a redist_map object A vector with a string for each measure desired. "PolsbyPopper", "Schwartzberg", "LengthWidth", "ConvexHull", "Reock", "BoyceClark", "FryerHolden", "EdgesRemoved", "FracKept", and "logSpanningTree" are implemented. Defaults to "PolsbyPopper". Use "all" to return all implemented measures. a redist_plans object passed on to redist.compactness A SpatialPolygonsDataFrame or sf object. Required unless "EdgesRemoved" and "logSpanningTree" with adjacency provided. A numeric vector (if only one map) or matrix with one row for each precinct and one column for each map. Required. A numeric vector with the population for every observation. Is only necessary when "FryerHolden" is used for measure. Defaults to NULL. A zero-indexed adjacency list. Only used for "PolsbyPopper", EdgesRemoved" and "logSpanningTree". Created with redist.adjacency if not supplied and needed. Default is NULL. A numeric to specify draw number. Defaults to 1 if only one map provided and the column number if multiple maps given. Can also take a factor input, which will become the draw column in the output if its length matches the number of entries in plans. If the plans input is a redist_plans object, it extracts the draw identifier. Number of cores to use for parallel computing. Default is 1. A numeric vector from 1:ncounties corresponding to counties. Required for "logSpanningTree". a number, indicating the CRS to project the shapefile to if it is latitude-longitude based. Set to FALSE to avoid planarizing. Boolean, whether to run Polsby Popper and Schwartzberg using Rcpp. It has a higher upfront cost, but quickly becomes faster. Becomes TRUE if ncol(district_membership > 8) and not manually set. it checks for an Rds, if no rds exists at the path, it creates an rds with borders and saves it. This can be created in advance with redist.prep.polsbypopper. A dataframe output from redist.prep.polsbypopper

## Value

A tibble with a column that specifies the district, a column for each specified measure, and a column that specifies the map number.

## Details

This function computes specified compactness scores for a map. If there is more than one shape specified for a single district, it combines them, if necessary, and computes one score for each district.

Polsby-Popper is computed as $$\frac{4*\pi*A(d)}{P(d)^2}$$ where A is the area function, the district is d, and P is the perimeter function. All values are between 0 and 1, where larger values are more compact.

Schwartzberg is computed as $$\frac{P(d)}{2*\pi*\sqrt{\frac{A(d)}{\pi}}}$$ where A is the area function, the district is d, and P is the perimeter function. All values are between 0 and 1, where larger values are more compact.

The Length Width ratio is computed as $$\frac{length}{width}$$ where length is the shorter of the maximum x distance and the maximum y distance. Width is the longer of the two values. All values are between 0 and 1, where larger values are more compact.

The Convex Hull score is computed as $$\frac{A(d)}{A(CVH)}$$ where A is the area function, d is the district, and CVH is the convex hull of the district. All values are between 0 and 1, where larger values are more compact.

The Reock score is computed as $$\frac{A(d)}{A(MBC)}$$ where A is the area function, d is the district, and MBC is the minimum bounding circle of the district. All values are between 0 and 1, where larger values are more compact.

The Boyce Clark Index is computed as $$1 - \sum_{1}^{16}\{\frac{|\frac{r_i}{\sum_ir_i}*100-6.25 |\}}{200}$$. The $$r_i$$ are the distances of the 16 radii computed from the geometric centroid of the shape to the most outward point of the shape that intersects the radii, if the centroid is contained within the shape. If the centroid lies outside of the shape, a point on the surface is used, which will naturally incur a penalty to the score. All values are between 0 and 1, where larger values are more compact.

The Fryer Holden score for each district is computed with $$Pop\odot D(precinct)^2$$, where $$Pop$$ is the population product matrix. Each element is the product of the ith and jth precinct's populations. D represents the distance, where the matrix is the distance between each precinct. To fully compute this index, for any map, the sum of these values should be used as the numerator. The denominator can be calculated from the full enumeration of districts as the smallest calculated numerator. This produces very large numbers, where smaller values are more compact.

The log spanning tree measure is the logarithm of the product of the number of spanning trees which can be drawn on each district.

The edges removed measure is number of edges removed from the underlying adjacency graph. A smaller number of edges removed is more compact.

The fraction kept measure is the fraction of edges that were not removed from the underlying adjacency graph. This takes values 0 - 1, where 1 is more compact.

## References

Boyce, R., & Clark, W. 1964. The Concept of Shape in Geography. Geographical Review, 54(4), 561-572.

Cox, E. 1927. A Method of Assigning Numerical and Percentage Values to the Degree of Roundness of Sand Grains. Journal of Paleontology, 1(3), 179-183.

Fryer R, Holden R. 2011. Measuring the Compactness of Political Districting Plans. Journal of Law and Economics.

Harris, Curtis C. 1964. “A scientific method of districting”. Behavioral Science 3(9), 219–225.

Maceachren, A. 1985. Compactness of Geographic Shape: Comparison and Evaluation of Measures. Geografiska Annaler. Series B, Human Geography, 67(1), 53-67.

Polsby, Daniel D., and Robert D. Popper. 1991. “The Third Criterion: Compactness as a procedural safeguard against partisan gerrymandering.” Yale Law & Policy Review 9 (2): 301–353.

Reock, E. 1961. A Note: Measuring Compactness as a Requirement of Legislative Apportionment. Midwest Journal of Political Science, 5(1), 70-74.

Schwartzberg, Joseph E. 1966. Reapportionment, Gerrymanders, and the Notion of Compactness. Minnesota Law Review. 1701.

## Examples

data(fl25) data(fl25_enum) plans_05 <- fl25_enum$plans[, fl25_enum$pop_dev <= 0.05] redist.compactness(shp = fl25, plans = plans_05[,1:3], measure = c('PolsbyPopper', 'EdgesRemoved'))
#> Linking to GEOS 3.8.1, GDAL 3.2.1, PROJ 7.2.1
#> Linking to liblwgeom 3.0.0beta1 r16016, GEOS 3.8.1, PROJ 7.2.1
#> # A tibble: 9 × 4 #> district PolsbyPopper EdgesRemoved draw #> <int> <dbl> <dbl> <dbl> #> 1 1 0.397 32 1 #> 2 2 0.246 32 1 #> 3 3 0.235 32 1 #> 4 1 0.284 44 2 #> 5 2 0.127 44 2 #> 6 3 0.235 44 2 #> 7 1 0.371 32 3 #> 8 2 0.234 32 3 #> 9 3 0.207 32 3